On Laguerre Isotropic Hypersurfaces

TL;DR

The paper investigates Laguerre isotropic hypersurfaces in Euclidean space, defined by vanishing Laguerre form and constant Laguerre tensor eigenvalues $\lambda\ge0$. It proves a rigidity result: if such a hypersurface is parametrized by lines of curvature, then $\lambda=0$ and the hypersurface is $L$-isoparametric, Laguerre-equivalent to a Dupin-type family; it further establishes that no curvature-line parametrization exists for $\lambda>0$ and provides a sharp bound $0<\rho^{2}<\frac{1}{2\lambda}$, where $\rho$ is a curvature-radii function. The work also shows that when $\lambda=0$ the $L$-isotropic and $L$-isoparametric conditions characterize hypersurfaces Laguerre-equivalent to the known Dupin example, thereby reducing the classification to a canonical Laguerre-equivalent family. Together, these results advance the understanding of Laguerre-invariant hypersurface geometry and clarify the rigidity and structure of isotropic configurations.

Abstract

We study Laguerre isotropic hypersurfaces in the Euclidean space, which are hypersurfaces whose Laguerre form is zero and the eigenvalues of the Laguerre tensor are constant and equal to $λ\geq 0$. We prove a rigidity theorem for the L-isotropic hypersurfaces parametrized by lines of curvature. Moreover, we study the hypersurfaces that are L-isotropic and L-isoparametric simultaneously and we show that for such a hypersurface $λ=0$. We obtain necessary conditions for the existence of L-isotropic hypersurfaces with $λ> 0$ and we prove that a certain function, determined by the radii of curvature of the hypersurface, is bounded above by ${1}/{2λ}$.

Theorems & Definitions (39)

• Proposition 1
• Theorem 1
• Proposition 2
• Theorem 2
• Definition 1
• Remark 3
• Proposition 3
• Definition 2
• Proposition 4
• Corollary 4